Exchange rates aren't a good way of measuring a currency, even though they would seem like one. The fact is that currency valuations are usually tied to the interest rates you can earn in banks of that currency. The ECB (European Central Bank) has always kept interest rates high. That encourages people to change dollars into euros and deposit them in European banks which drives up the price of the euro in foreign exchange markets. However, high interest rates also discourage the investment in businesses that drives an economy. So, one can't say that the euro's strength benefits its people.
And that's what you need to do: to look at how a currency is serving the people under it. The Federal Reserve has proven over decades that it can effectively fight both booms and recessions. In a recession, one needs to expand the money supply. That makes it easier for borrowing and therefore investing and spending - exactly what we're seeing missing from our current situation. People have stopped borrowing, investing, and spending because of the high cost of it (compared to boom times) and so by making it easier to do those things we can mitigate some of the pain of the recession and make the whole recession shorter as we more quickly encourage business to return to normal.*
The ECB has lowered interest rates, but the ECB's policy has always been to target inflation and not mitigate recession. Partly this comes from differences between Europe and America. Europe has had many currencies in the past that just free-fell. Inflation was terrible, currency unstable, etc. Likewise, governments have large social programs in Europe and high unemployment is seen as a way of life rather than a disaster. For example, in Germany 8% unemployment is amazing. In the US, we call for politicians' heads when it hits that level. But European countries are more capable of handling such unemployment without huge upheaval because the social programs keep people at bay and without as much worry about a paycheck.
*This is the consensus of mainstream economics in the same way that the existence of global warming is the consensus of mainstream climate scientists. Yes, you can always find those who will disagree with anything.